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# An application of the inequality for modified Poisson kernel

Journal of Inequalities and Applications20162016:24

https://doi.org/10.1186/s13660-016-0959-6

• Received: 14 September 2015
• Accepted: 4 January 2016
• Published:

## Abstract

As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.

## Keywords

• growth property
• Dirichlet problem
• modified Poisson kernel

## 1 Introduction and results

Let $$\mathbf{R}^{n}$$ ($$n\geq2$$) be the n-dimensional Euclidean space. The boundary and the closure of a set E in $$\mathbf{R}^{n}$$ are denoted by ∂E and , respectively. The Euclidean distance of two points P and Q in $$\mathbf{R}^{n}$$ is denoted by $$\vert P-Q\vert$$. Especially, $$\vert P\vert$$ denotes the distance of two points P and O in $$\mathbf{R}^{n}$$, where O is the origin in $$\mathbf{R}^{n}$$.

We introduce a system of spherical coordinates $$(r,\Theta)$$, $$\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})$$, in $$\mathbf{R}^{n}$$ which are related to Cartesian coordinates $$(x_{1},x_{2},\ldots,x_{n-1},x_{n})$$ by $$x_{n}=r\cos\theta_{1}$$.

Let $$B(P,r)$$ denote the open ball with center at P and radius r (>0) in $$\mathbf{R}^{n}$$. The unit sphere and the upper half unit sphere in $$\mathbf{R}^{n}$$ are denoted by $$\mathbf{S}^{n-1}$$ and $$\mathbf{S}_{+}^{n-1}$$, respectively. The surface area $$2\pi^{n/2}\{\Gamma(n/2)\}^{-1}$$ of $$\mathbf{S}^{n-1}$$ is denoted $$w_{n}$$. Let $$\Omega\subset\mathbf{S}^{n-1}$$, a point $$(1,\Theta)$$ and the set $$\{\Theta; (1,\Theta)\in\Omega\}$$ are denoted Θ and Ω, respectively. For two sets $$\Lambda\subset\mathbf{R}_{+}$$ and $$\Omega\subset\mathbf{S}^{n-1}$$, we denote $$\Lambda\times\Omega=\{ (r,\Theta)\in\mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Omega\}$$, where $$\mathbf{R}_{+}$$ is the set of all positive real numbers.

For the set $$\Omega\subset\mathbf{S}^{n-1}$$, we denote the set $$\mathbf{ R}_{+}\times\Omega$$ in $$\mathbf{R}^{n}$$ by $$C_{n}(\Omega)$$, which is called a cone. For the set $$I\subset \mathbf{R}$$, the sets $$I\times\Omega$$ and $$I\times\partial{\Omega}$$ are denoted $$C_{n}(\Omega;I)$$ and $$S_{n}(\Omega;I)$$, respectively, where R is the set of all real numbers. Especially, the set $$S_{n}(\Omega; \mathbf{R}_{+})$$ is denoted $$S_{n}(\Omega)$$.

Given a continuous function f on $$S_{n}(\Omega)$$, we say that h is a solution of the Dirichlet problem in $$C_{n}(\Omega)$$ with f, if h is a harmonic function in $$C_{n}(\Omega)$$ and
$$\lim_{P\rightarrow Q\in S_{n}(\Omega), P\in C_{n}(\Omega)}h(P)=f(Q).$$
Let $$\Omega\subset\mathbf{S}^{n-1}$$ and $$\Delta^{*}$$ be a Laplace-Beltrami on the unit sphere. Consider the Dirichlet problem (see, e.g. , p.41)
\begin{aligned}& \Delta^{*}\varphi(\Theta)+\lambda\varphi(\Theta)=0 \quad \text{in } \Omega, \\& \varphi(\Theta)=0 \quad \text{in } \partial{\Omega}. \end{aligned}
We denote the non-decreasing sequence of positive eigenvalues of it, repeating accordingly to their multiplicities, and the corresponding eigenfunctions are denoted, respectively, by $$\{\lambda_{i}\}_{i=1}^{\infty}$$ and $$\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}$$. Especially, we denote the least positive eigenvalue of it $$\lambda_{1}$$ and the normalized positive eigenfunction to $$\lambda_{1}$$ $$\varphi_{1}(\Theta)$$. In the sequel, for the sake of brevity, we shall write λ and φ instead of $$\lambda_{1}$$ and $$\varphi_{1}$$, respectively.
The set of sequential eigenfunctions corresponding to the same value of $$\{\lambda_{i}\}_{i=1}^{\infty}$$ in the sequence $$\{\varphi_{i}(\Theta)\}_{i=1}^{\infty}$$ makes an orthonormal basis for the eigenspace of the eigenvalue $$\lambda_{i}$$. Hence for each $$\Omega\subset S^{n-1}$$ there is a sequence $$\{k_{j}\}$$ of positive integers such that $$k_{1}=1$$, $$\lambda_{k_{j}}<\lambda_{k_{j+1}}$$, $$\lambda_{k_{j}}=\lambda_{k_{j}+1}=\lambda_{k_{j}+2}=\cdots=\lambda_{k_{j+1}-1}$$ and $$\{\varphi_{k_{j}},\varphi_{k_{j}+1},\ldots,\varphi_{k_{j+1}-1}\}$$ is an orthonormal basis for the eigenspace of the eigenvalue $$\{\lambda_{k_{j}}\}_{j=1}^{\infty}$$. By $$I_{\Omega}(k_{m})$$ we denote the set of all positive integers less than $$\{k_{m}\}_{m=1}^{\infty}$$. In spite of the fact
$$I_{\Omega}(k_{1})=\varnothing,$$
the summation over $$I_{\Omega}(k_{1})$$ of a function $$S(k)$$ of a variable k will be used by promising
$$\sum_{k\in I_{\Omega}(k_{1})}S(k)=0.$$
If we denote the solutions of the equation
$$t^{2}+(n-2)t-\lambda_{i}=0\quad (i=1,2,3,\ldots)$$
by $$\aleph_{i}^{+}$$ and $$\aleph_{i}^{-}$$, then the functions
$$r^{\aleph_{i}^{\pm}}\varphi_{i}(\Theta) \quad (i=1,2,3,\ldots)$$
are harmonic functions in $$C_{n}(\Omega)$$ and vanish on $$S_{n}(\Omega)$$.
Let $$G_{\Omega}(P,Q)$$ be the Green function of $$C_{n}(\Omega)$$ for any $$P=(r,\Theta)\in C_{n}(\Omega)$$ and any $$Q=(t,\Phi)\in C_{n}(\Omega)$$. Then the Poisson kernel in $$C_{n}(\Omega)$$ can be defined by
$$PI_{\Omega}(P,Q)=\frac{1}{c_{n}}\frac{\partial}{\partial n_{Q}}G_{\Omega}(P,Q),$$
where $$P\in C_{n}(\Omega)$$, $$Q\in S_{n}(\Omega)$$, $${\partial}/{\partial n_{Q}}$$ denotes the differentiation at Q along the inward normal into $$C_{n}(\Omega)$$ and
$$c_{n}=\textstyle\begin{cases} 2\pi & \mbox{if } n=2, \\ (n-2)w_{n} & \mbox{if } n\geq3. \end{cases}$$
Let $$F(\Theta)$$ be a function defined in Ω. We denote $$N_{i}(F)$$ by
$$\int_{\Omega}F(\Theta)\varphi_{i}(\Theta)\,d\Omega,$$
when it exists.
For any two points $$P=(r,\Theta)$$ and $$Q=(t,\Phi)$$ in $$C_{n}(\Omega)$$ and $$S_{n}(\Omega)$$, respectively, we define
$$\widetilde{K}_{\Omega}^{m}(P,Q)=\textstyle\begin{cases} 0 & \mbox{if } 0< t< 1, \\ K_{\Omega}^{m}(P,Q) & \mbox{if } 1\leq t< \infty, \end{cases}$$
where m is a non-negative integer and
$$K_{\Omega}^{m}(P,Q)=\sum_{i\in I_{k_{m+1}}}2^{\aleph _{i}^{+}+n-1}N_{i} \bigl(PI_{\Omega}\bigl((1,\Theta),(2,\Phi)\bigr)\bigr)r^{\aleph _{i}^{+}}t^{-\aleph_{i}^{+}-n+1} \varphi_{i}(\Theta).$$
To obtain the solution of the Dirichlet problem in a cone, as in [1, 3, 4], we use the modified Poisson kernel defined by
$$PI_{\Omega}^{m}(P,Q)=PI_{\Omega}(P,Q)- \widetilde{K}_{\Omega}^{m}(P,Q),$$
where $$P\in C_{n}(\Omega)$$ and $$Q\in S_{n}(\Omega)$$, which has the following estimates (see ):
$$\bigl\vert PI_{\Omega}(P,Q)-K_{\Omega}^{m}(P,Q) \bigr\vert \leq M(2r)^{\aleph_{k_{m+1}}^{+}}t^{-\aleph_{k_{m+1}}^{+}-n+1}$$
(1)
for any $$P=(r,\Theta)\in C_{n}(\Omega)$$ and any $$Q=(t,\Phi)\in S_{n}(\Omega)$$ satisfying $$0<\frac{r}{t}<\frac{1}{2}$$, where M is a constant independent of P, Q, and m. For the construction and applications of a modified Green function in a half space, we refer the reader to the paper by Qiao (see ).
Write
$$U_{\Omega}^{m}[f](P)= \int_{S_{n}(\Omega)}PI_{\Omega}^{m}(P,Q)f(Q)\,d \sigma_{Q},$$
where $$f(Q)$$ is a continuous function on $$\partial C_{n}(\Omega)$$ and $$d\sigma_{Q}$$ the $$(n-1)$$-dimensional volume elements induced by the Euclidean metric on $$\partial{C_{n}(\Omega)}$$.

Recently, Qiao and Deng (cf. ) gave the solution of the Dirichlet problem in a cone. Applications of modified Poisson kernel with respect to the Schrödinger operator, we refer the reader to the papers by Huang and Ychussie (see ) and Li and Ychussie (see ).

### Theorem A

If $$\Omega+\aleph^{+}-1>0$$, $$\Omega-n+1\leq\aleph_{k_{m+1}}^{+}<\Omega-n+2$$ and $$f(Q)$$ ($$Q=(t,\Phi )$$) is a continuous function on $$\partial{C_{n}(\Omega)}$$ satisfying
$$\int_{S_{n}(\Omega)}\frac{\vert f(Q)\vert }{1+t^{\Omega}}\,d\sigma_{Q}< \infty,$$
(2)
then the function $$U_{\Omega}^{m}[f](P)$$ is a solution of the Dirichlet problem in $$C_{n}(\Omega)$$ with f and
$$\lim_{r\rightarrow\infty, P=(r,\Theta)\in C_{n}(\Omega)}r^{n-\Omega-1}\varphi^{n-1}(\Theta) U_{\Omega}^{m}[f](P)=0.$$

Furthermore, Qiao and Deng (cf. ) supplemented the above result and proved the following.

### Theorem B

Let $$0< p<\infty$$, $$\gamma>(-\aleph^{+}-n+2)p+n-1$$ and
$$\frac{\gamma-n+1}{p}< \aleph_{k_{m+1}}^{+}< \frac{\gamma-n+1}{p}+1.$$
If $$f(Q)$$ ($$Q=(t,\Phi)$$) is a continuous function on $$S_{n}(\Omega)$$ satisfying
$$\int_{S_{n}(\Omega)}\frac {\vert f(Q)\vert ^{p}}{1+t^{\gamma}}\,d\sigma_{Q}< \infty,$$
(3)
then the function $$U_{\Omega}^{m}[f](P)$$ satisfies
$$\lim_{r\rightarrow\infty, P=(r,\Theta)\in C_{n}(\Omega)}r^{\frac{n-\gamma-1}{p}}\varphi^{n-1}(\Theta) U_{\Omega}^{m}[f](P)=0.$$
It is natural to ask if the continuous function u satisfying (2) and (3) can be replaced by arbitrary continuous function? In this paper, we shall give an affirmative answer to this question. To do this, we first construct a modified Poisson kernel. Let $$\phi(l)$$ be a positive function of $$l\geq1$$ satisfying
$$2^{\aleph^{+}}\phi(1)=1.$$
Denote the set
$$\bigl\{ l\geq1;-\aleph_{k_{i}}^{+}\log2=\log \bigl(l^{n-1}\phi(l)\bigr)\bigr\}$$
by $$\pi_{\Omega}(\phi,i)$$. Then $$1\in\pi_{\Omega}(\phi,i)$$. When there is an integer N such that $$\pi_{\Omega}(\phi,N)\neq\Phi$$ and $$\pi_{\Omega}(\phi,N+1)= \Phi$$, denote
$$J_{\Omega}(\phi)=\{i;1\leq i\leq N\}$$
of integers. Otherwise, denote the set of all positive integers by $$J_{\Omega}(\phi)$$. Let $$l(i)=l_{\Omega}(\phi,i+1)$$ be the minimum elements l in $$\pi _{\Omega}(\phi,i)$$ for each $$i\in J_{\Omega}(\phi)$$. In the former case, we put $$l{(N+1)}=\infty$$. Then $$l(1)=1$$. The kernel function $$\widetilde{K}_{\Omega}^{\phi}(P,Q)$$ is defined by
$$\widetilde{K}_{\Omega}^{\phi}(P,Q)=\textstyle\begin{cases} 0 & \mbox{if } 0< t< 1, \\ K_{\Omega}^{i}(P,Q) & \mbox{if } l(i)\leq t< l(i+2) \text{ and } i\in J_{\Omega}(\phi), \end{cases}$$
where $$P\in C_{n}(\Omega)$$ and $$Q=(t,\Phi)\in S_{n}(\Omega)$$.
The generalized Poisson kernel $$P_{\Omega}^{\phi}(P,Q)$$ is defined by
$$PI_{\Omega}^{\phi}(P,Q)=PI_{\Omega}(P,Q)- \widetilde{K}_{\Omega}^{\phi}(P,Q),$$
where $$P\in C_{n}(\Omega)$$ and $$Q\in S_{n}(\Omega)$$.

As an application of the inequality (1) and the generalized Poisson kernel $$PI_{\Omega}^{\phi}(P,Q)$$, we have the following.

### Theorem

Let $$g(Q)$$ be a continuous function on $$S_{n}(\Omega)$$. Then there is a positive continuous function $$\phi_{g}(l)$$ of $$l\geq0$$ depending on g such that
$$H_{\Omega}^{\phi_{g}}(P)= \int_{S_{n}(\Omega)}PI_{\Omega}^{\phi _{g}}(P,Q)g(Q)\,d \sigma_{Q}$$
is a solution of the Dirichlet problem in $$C_{n}(\Omega)$$ with g.

## 2 Lemmas

### Lemma 1

Let $$\phi(l)$$ be a positive continuous function of $$l\geq1$$ satisfying
$$\phi(1)=2^{-\aleph^{+}}.$$
Then
$$\bigl\vert PI_{\Omega}(P,Q)-\widetilde{K}_{\Omega}^{\phi}(P,Q) \bigr\vert \leq M \phi(l)$$
for any $$P=(r,\Theta)\in C_{n}(\Omega)$$ and any $$Q=(t,\Phi)\in S_{n}(\Omega)$$ satisfying
$$t>\max\{1,4r\}.$$
(4)

### Proof

We can choose two points $$P=(r,\Theta)\in C_{n}(\Omega)$$ and $$Q=(t,\Phi)\in S_{n}(\Omega)$$, which satisfies (4). Moreover, we also can choose an integer $$i=i(P,Q)\in J_{\Omega}(\phi)$$ such that
$$l(i-1)\leq t< l(i).$$
(5)
Then
$$\widetilde{K}_{\Omega}^{\phi}(P,Q)=K_{\Omega}^{i-1}(P,Q).$$
Hence we have from (1), (4), and (5) that
$$\bigl\vert PI_{\Omega}(P,Q)-\widetilde{K}_{\Omega}^{\phi}(P,Q) \bigr\vert \leq M 2^{-\aleph _{k_{i}}^{+}} \leq M \phi(l),$$
which is the conclusion. □

### Lemma 2

(See )

Let $$g(Q)$$ be a continuous function on $$\partial{C_{n}(\Omega)}$$ and $$V(P,Q)$$ be a locally integrable function on $$\partial{C_{n}(\Omega)}$$ for any fixed $$P\in C_{n}(\Omega)$$, where $$Q\in \partial{C_{n}(\Omega)}$$. Define
$$W(P,Q)=PI_{\Omega}(P,Q)-V(P,Q)$$
for any $$P\in C_{n}(\Omega)$$ and any $$Q\in\partial{C_{n}(\Omega)}$$.

Suppose that the following two conditions are satisfied:

(I) For any $$Q'\in\partial{C_{n}(\Omega)}$$ and any $$\epsilon>0$$, there exists a neighborhood $$B(Q')$$ of $$Q'$$ such that
$$\int_{S_{n}(\Omega;[R,\infty))}\bigl\vert W(P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \,d\sigma_{Q}< \epsilon$$
(6)
for any $$P=(r,\Theta)\in C_{n}(\Omega)\cap B(Q')$$, where R is a positive real number.
(II) For any $$Q'\in\partial{C_{n}(\Omega)}$$, we have
$$\limsup_{P\rightarrow Q', P\in C_{n}(\Omega)} \int_{S_{n}(\Omega ;(0,R))}\bigl\vert V(P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \,d\sigma_{Q}=0$$
(7)
for any positive real number R.
Then
$$\limsup_{P\rightarrow Q', P\in C_{n}(\Omega)} \int_{S_{n}(\Omega )}W(P,Q)u\bigl(Q'\bigr)\,d\sigma_{Q}\leq u(Q)$$
for any $$Q'\in\partial{C_{n}(\Omega)}$$.

## 3 Proof of Theorem

Take a positive continuous function $$\phi(l)$$ ($$l\geq1$$) such that
$$\phi(1)2^{\aleph^{+}}=1$$
(8)
and
$$\phi(l) \int_{\partial\Omega}\bigl\vert g(l,\Phi)\bigr\vert \,d \sigma_{\Phi}\leq\frac{L}{l^{n}}$$
for $$l>1$$, where
$$L= 2^{-\aleph^{+}} \int_{\partial\Omega}\bigl\vert g(1,\Phi)\bigr\vert \,d \sigma_{\Phi}.$$
For any fixed $$P=(r,\Theta)\in C_{n}(\Omega)$$, we can choose a number R satisfying $$R>\max\{1,4r\}$$. Then we see from Lemma 1 that
\begin{aligned} & \int_{S_{n}(\Omega;(R,\infty))}\bigl\vert PI_{\Omega}^{\phi_{g}}(P,Q) \bigr\vert \bigl\vert g(Q)\bigr\vert \,d\sigma _{Q} \\ &\quad \leq M \int_{R}^{\infty} \biggl( \int_{\partial\Omega}\bigl\vert g(1,\Phi)\bigr\vert \,d \sigma_{\Phi} \biggr)\phi(l)l^{n-2}\,dl \\ &\quad \leq\quad ML \int_{R}^{\infty} l^{-2}\,dl \\ &\quad < \infty. \end{aligned}
(9)
Obviously, we have
$$\int_{S_{n}(\Omega;(0,R))}\bigl\vert PI_{\Omega}^{\phi_{g}}(P,Q) \bigr\vert \bigl\vert g(Q)\bigr\vert \,d\sigma _{Q}< \infty,$$
which gives
$$\int_{S_{n}(\Omega)}\bigl\vert PI_{\Omega}^{\phi_{g}}(P,Q) \bigr\vert \bigl\vert g(Q)\bigr\vert \,d\sigma_{Q}< \infty.$$

To see that $$H_{\Omega}^{\phi_{g}}(P)$$ is a harmonic function in $$C_{n}(\Omega)$$, we remark that $$H_{\Omega}^{\phi_{g}}(P)$$ satisfies the locally mean-valued property by Fubini’s theorem.

Finally we shall show that
$$\lim_{P\in C_{n}(\Omega),P\rightarrow Q'}H_{\Omega}^{\phi_{g}}(P)=g \bigl(Q'\bigr)$$
for any $$Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}$$. Set
$$V(P,Q)=\widetilde{K}_{\Omega}^{\phi_{g}}(P,Q)$$
in Lemma 2, which is locally integrable on $$S_{n}(\Omega)$$ for any fixed $$P\in C_{n}(\Omega)$$. Then we apply Lemma 2 to $$g(Q)$$ and $$-g(Q)$$.

For any $$\epsilon>0$$ and a positive number δ, by (9) we can choose a number R ($$>\max\{1,2(t'+\delta)\}$$) such that (6) holds, where $$P\in C_{n}(\Omega)\cap B(Q',\delta)$$.

Since
$$\lim_{\Theta\rightarrow \Phi'}\varphi_{i}(\Theta)=0\quad (i=1,2,3\ldots)$$
as $$P=(r,\Theta)\rightarrow Q'=(t',\Phi')\in S_{n}(\Omega)$$, we have
$$\lim_{P\in C_{n}(\Omega),P\rightarrow Q'} \widetilde{K}_{\Omega}^{\phi_{g}}(P,Q)=0,$$
where $$Q\in S_{n}(\Omega)$$ and $$Q'\in S_{n}(\Omega)$$. Then (7) holds.

Thus we complete the proof of Theorem.

## Declarations

### Acknowledgements

The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions. 